An Efficient Approximate Algorithm for Winner Determination in Combinatorial Auctions

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1 An Efficient Approximate Algorithm for Winner Determination in Combinatorial Auctions Yuko Sakurai, Makoto Yokoo, and Koji Kamei NTT Communication Science Laboratories, 2-4 Hikaridai, Seika-cho, Soraku-gun, Kyoto, , Japan Introduction Auctions on the Internet have become especially popular in Electronic Commerce. Amongthe various studies on the Internet auctions, those on combinatorial auctions have lately attracted considerable attention. Combinatorial auctions simultaneously sell multiple items with interdependent values and allow the bidders to bid on any combination of items. Therefore, they tend to increase the buyers utilities and the seller s revenue. On the other hand, determiningwinners in combinatorial auctions intended to maximize the sum of the bids with disjoint sets of items is a complicated constraint optimization problem and shown to be NP-complete [4]. This problem has recently attracted the interest of the AI community as an application of search techniques [,5]. Although the previous methods have significantly improved the efficiency of the optimal winner determination algorithms, to solve large-scale problems, we eventually need to give up the idea of achieving the optimality of the obtained allocation and try to find a semi-optimal solution within a reasonable amount of time. In this paper, we introduce limited discrepancy search (LDS) techniques [2] to limit the search efforts to the part where good solutions are likely to exist. Experimental evaluations using various problem settings show that our algorithm finds the allocations that are very close to the optimal solutions (better than 95%) very quickly (in about % of the runningtime) and can be extended to large-scale problem instances. 2 Preliminaries 2. Winner Determination Problem A winner determination problem can be formulated as follows [4,5]. Let G = {, 2,,m} be the set of items to be auctioned and A be a set of bidders. Suppose a bidder i A can submit any bid b i (S) for any combination of items S G. Let b(s) be the the maximal bid for the bundle S, thatis, b(s) = R. Dechter (Ed.): CP 2000, LNCS 894, pp , c Springer-Verlag Berlin Heidelberg 2000

2 550Yuko Sakurai et al. Fig.. Search Tree Fig. 2. Discrepancies of Nodes max i A b i (S), and χ be the set of possible allocations, that is, χ = {S G S S = for every S, S χ}. Then, the goal is to find an allocation χ such that argmax χ b(s). () S χ 2.2 Problem Formalization as Search Problem We basically follow the problem formalization used in [5]. An example of a search tree is presented in Figure. Each node is associated with a bid, and a path from the root node to any other node represents one (partial) allocation, which consists of a sequence of disjoint bids. The child nodes of each node are limited to the nodes that include the item with the smallest index amongthe items that are not yet on the path but that do not include items that are already on the path. In [5], a search algorithm called IDA* [3] is used for searchingthis tree. We introduce a special data structure called bins [] to find quickly the children of each nodes. More specifically, we sort bids into bins, where a bin B j contains all bids where j is the smallest index amongthe items in a bid. The followingheuristic function h is used to estimate the possible maximal revenue for the items not yet allocated on the path in [5]. h = k {unallocated items} r(k), where r(k) = b(s) max {S;k S} S (2) If we re-calculate h in IDA* whenever a bid is appended to the path, h becomes more accurate, but this re-calculation requires significant overhead. We found that this re-calculation is necessary in IDA* to decrease the number of visited nodes. On the other hand, we can avoid this re-calculation since LDS is less sensitive to the accuracy of heuristic evaluations. We use an evaluation function f(n) = g(n) + h(n) to estimate the best (highest) price of the solutions obtained from node n. g(n) is calculated as the sum of the bids on the path appended to n. h(n) is a heuristic function defined as (2). We use f(n) used for orderingsiblingnodes and for pruning.

3 An Efficient Approximate Algorithm for Winner Determination 55. If stack is empty, then terminate the algorithm, otherwise, pop list from stack. 2. If list is empty, go to, otherwise, set n c to the first node in list, andremoven c. 3. If dis(n c) >D max, thengoto. 4. If f(n c) f max, thengoto. 5. If n c is a leaf node, then record the current path as a best solution, set f max to f(n c), set n c to the first node in list, removen c from list, and go to Expand n c. push list to stack, setn c to the best child, and set list to the rest of the children sorted by f, goto3. Fig. 3. Pseudo Code of LDS Algorithm 2.3 Limited Discrepancy Search (LDS) The original LDS algorithm was developed for searching a binary tree. As shown in [2], there are several alternative ways to modify LDS for a non-binary search tree. We define the discrepancy of node n (represented as dis(n)) as follows. The rank of node n (represented as rank(n)) is defined as the order amongits siblings, i.e., the rank of the best child node is 0, the second-best node is, and so on. By representingthe parent node as n p, dis(n) is defined as dis(n p )+rank(n), where the discrepancy of the root node dis(root) = 0. Figure 2 shows the discrepancies of nodes of a search tree in Figure. Figure 3 shows the pseudo code of the LDS algorithm. We store a list of nodes, which are siblings sorted by f, intostack. f max represents the highest price of the solutions found so far, and D max is the maximal number of allowed discrepancies. Initially, the stack contains a list of the root node, and f max is 0. 3 Experiments We ran a series of experiments for several problem settings used in previous works [,5] on a workstation (333 MHz Sun UltraSparc IIi with 52 MB) with a program written in C++. Due to space limitations, we only show the graphs of experiments on the followingtwo distributions used in [5]. Let M be the number of items and N be the, where bids are different from one another. The trends of the obtained results on other problem settings used in [] were also similar to the results described here. Random Distribution The number of items in each bid is randomly chosen from [,M], and items included in a bid are also randomly chosen. The price is randomly chosen from [0, ]. Uniform Distribution The number of items is set to a constant k (we set k = 3). The items and the price are chosen in the same manner as above. We generated 0 problem instances and calculated the average of them. To compare the results of LDS without re-calculation and the result of IDA* with re-calculation, we present () quality of obtained solution, i.e., the ratio of the

4 552 Yuko Sakurai et al. Average Number of Visited Nodes e+08 e+07 e () Solution Quality IDA* without re-calc. IDA* with re-calc (3) Visited Nodes Average Time Average Social Surplus IDA* without re-calc. IDA* with re-calc (2) Running Time 8 6 IDA* (4) Social Surplus LDS without re-calculation LDS with re-calculation Time (sec) (5) Anytime Performance Fig. 4. Random Distribution sum of the winningbids in LDS to the sum in IDA*, (2) runningtime, (3) number of visited nodes, and (4) social surplus, i.e., the sum of the winning bids, while varyingthe number of submitted bids, where we limit the maximal discrepancy to i [0, 2] (represented as Di) inlds. Random (Figures 4): As a comparison, we also show the results of IDA* without re-calculation. Figures 4 show the running time and the number of visited nodes of IDA* increase exponentially (note that the y-axis is logscaled), while those of LDS increase rather slowly. LDS with D2 remains better than 95% even for lage-scale problem instances. LDS can be considered an anytime algorithm. This property is desirable for winner determination problems. We illustrate (5) a comparison of the average anytime performance of LDS without re-calculation and with recalculation. We generated 0 instances where M = 400 and N = 500 and

5 An Efficient Approximate Algorithm for Winner Determination () Solution Quality Average Time Fig. 5. Uniform Distribution IDA* (2) Running Time gradually increased the maximal discrepancy (one-by-one from D0). This result shows, if we do not have enough time, it is reasonable to use the LDS algorithm without re-calculation to obtain a semi-optimal solution. Uniform (Figures 5): Figures 5 show the results where we set M = 00. IDA* can solve only problem instances where the is small within a reasonable amount of time. On the other hand, since LDS restricts the search space accordingto the maximal discrepancy, LDS can find about 95% of the optimal solution very quickly. 4 Conclusions Determiningthe winners in combinatorial auctions is a complicated constraint optimization problem. We have presented an approximate algorithm that is based on the LDS technique. Experimental evaluations performed on various problem sets showed that the LDS algorithm can find a semi-optimal solution (better than 95%) with a small amount of search effort (in about % of the runningtime) compared with the existingoptimal algorithm. References. Fujishima, Y., Leyton-Brown, K., and Shoham, Y.: Taming the Computational Complexity of Combinatorial Auctions: Optimal and Approximate Approaches. IJCAI-99 (999) 549, 550, Harvey,W.D.andGinsberg,M.L.:LimitedDiscrepancySearch.IJCAI-95 (995) 549, Korf, R. E.: Depth-first iterative deepening: an optimal admissible tree search. Artificial Intelligence 62 (993) Rothkopf, M. H., Pekeč, A., and Harstad, R. M.: Computationally Manageable Combinatorial Auctions. Management Science 44 (998) Sandholm, T.: An Algorithm for Optimal Winner Determination in Combinatorial Auction. IJCAI-99 (999) 549, 550, 55